This insight is used to extend the definition of exponentiation to include negative and fractional exponents

#a^(-x) = 1/(a^x)#

#a^(1/x) = x^(th)# root of #a# (sorry; I don't know how to get the symbols for this)

Also#(a^x)/(a^y) = a^(x-y)#
and#(a^x)^y = (a^y)^x = a^(xy)#

Logarithms:#log# with some specified base is a function.

Definition:
The value of #log_b(m) =# the value of #y# needed to make #b^y = m#
(Repeat that multiple times before going on. Come back to it as often as you need).

Example:#log_3(81) = 4# since #3^4 = 81#

Common log identities:#log_b(b) = 1#

#log_b(b^x) = x#

#log_b(c^x) = x log_b(c)#

#log_b(mn) = log_b(m) + log_b(n)#

#log_b(m/n) = log_b(m) - log_b(n)#

#log_b(m^n) = n log_b(m)#

Some special #log# notes:

Sometimes you may see #log (m)# written without a base specified; the convention in this case is that the base is #10#

Often you will see #ln(m)#; this is another form of the #log# function with a special base value, #e#; that is #ln(m) = log_e(m)#
where #e# is a special number (like #pi#) approximately equal to #2.72#; #e# has some special properties that make it useful in calculus.